11. Design and implement C/C++ Program to sort a given set of n integer elements using Merge Sort method and compute its time complexity. Run the program for varied values of n> 5000, and record the time taken to sort. Plot a graph of the time taken versus n. The elements can be read from a file or can be generated using the random number generator.
Step 1: Implement the Merge Sort Algorithm
Merge Sort is a divide-and-conquer algorithm that splits the array into values, sorts each half, and then merges the sorted values.
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
// Function to merge two sorted arrays
void merge(int arr[], int left, int mid, int right)
{
int i, j, k;
int n1 = mid - left + 1;
int n2 = right - mid;
int *L = (int *)malloc(n1 * sizeof(int));
int *R = (int *)malloc(n2 * sizeof(int));
for (i = 0; i < n1; i++)
L[i] = arr[left + i];
for (j = 0; j < n2; j++)
R[j] = arr[mid + 1 + j];
i = 0;
j = 0;
k = left;
while (i < n1 && j < n2)
{
if (L[i] <= R[j])
{
arr[k] = L[i];
i++;
}
else
{
arr[k] = R[j];
j++;
}
k++;
}
while (i < n1)
{
arr[k] = L[i];
i++;
k++;
}
while (j < n2)
{
arr[k] = R[j];
j++;
k++;
}
free(L);
free(R);
}
// Function to implement Merge Sort
void mergeSort(int arr[], int left, int right)
{
if (left < right)
{
int mid = left + (right - left) / 2;
mergeSort(arr, left, mid);
mergeSort(arr, mid + 1, right);
merge(arr, left, mid, right);
}
}
// Function to generate random integers
void generateRandomArray(int arr[], int n)
{
for (int i = 0; i < n; i++)
arr[i] = rand() % 100000; // Generate random integers between 0 and 99999
}
int main()
{
int n;
printf("Enter the number of elements: ");
scanf("%d", &n);
if (n <= 5000)
{
printf("Please enter a value greater than 5000\n");
return 1; // Exit if the number of elements is not greater than 5000
}
int *arr = (int *)malloc(n * sizeof(int));
if (arr == NULL)
{
printf("Memory allocation failed\n");
return 1; // Exit if memory allocation fails
}
generateRandomArray(arr, n);
// Repeat the sorting process multiple times to increase duration for timing
clock_t start = clock();
for (int i = 0; i < 1000; i++)
{
mergeSort(arr, 0, n - 1);
}
clock_t end = clock();
// Calculate the time taken for one iteration
double time_taken = ((double)(end - start)) / CLOCKS_PER_SEC / 1000.0;
printf("Time taken to sort %d elements: %f seconds\n", n, time_taken);
free(arr);
return 0;
}Step 2: Measure Time Taken
This program generates n random numbers, sorts them using the Merge Sort algorithm, and measures the time taken for the sorting process.
Step 3: Run the Program for Various Values of n
To collect data, run the program with different values of n greater than 5000, such as 6000, 7000, 8000, etc., and record the time taken for each.
Step 4: Plot the Results
You can use a graphing tool like Python with matplotlib to plot the results.
import matplotlib.pyplot as plt
# data collected (replace with actual data)
n_values = [6000, 7000, 8000, 9000, 10000, 11000, 12000, 13000, 15000]
time_taken = [0.000709, 0.000752, 0.000916, 0.001493, 0.001589, 0.002562, 0.001944, 0.002961, 0.003563] # Replace with actual times recorded
plt.plot(n_values, time_taken, marker='o')
plt.title('Merge Sort Time Complexity')
plt.xlabel('Number of Elements (n)')
plt.ylabel('Time taken (seconds)')
plt.grid(True)
plt.show()Enter number of elements: 6000
Time taken to sort 6000 elements: 0.000709 seconds
********************************************************************
Enter number of elements: 7000
Time taken to sort 7000 elements: 0.000752 seconds
********************************************************************
Enter number of elements: 8000
Time taken to sort 8000 elements: 0.000916 seconds
********************************************************************
Enter number of elements: 9000
Time taken to sort 9000 elements: 0.001493 seconds
********************************************************************
Enter number of elements: 10000
Time taken to sort 10000 elements: 0.001589 seconds
********************************************************************
Enter number of elements: 11000
Time taken to sort 11000 elements: 0.002562 seconds
********************************************************************
Enter number of elements: 12000
Time taken to sort 12000 elements: 0.001944 seconds
********************************************************************
Enter number of elements: 13000
Time taken to sort 13000 elements: 0.002961 seconds
********************************************************************
Enter number of elements: 15000
Time taken to sort 15000 elements: 0.003563 seconds